Numerical Solution of Two Transcendental Equations
نویسندگان
چکیده
This paper deals with the study of the transcendental equations: sin(j + v)/(s + v) = ±sin(i v)/(s v), where v = (s2 y2)'/2. These equations are obtained in the study of some boundary value problems for a modified biharmonic equation using the PapkovichFadle series. Some numerical solutions obtained with an iterative procedure are given. Introduction. It is well known that the governing equation of a bidimensional Stokes flow parallel to the x, y plane, in terms of the stream function ^(x, y), is the biharmonic equation „v 4T a4* a4* a4* . (0 V4* =-7 + —:—+-r = 0. V dx4 dx2dy2 dy4 The solution of Eq. (1) in a rectangle with homogeneous conditions on ¥ and on its normal derivative on two parallel sides and for sufficiently regular boundary conditions on the remaining two parallel sides is expressed using the PapkovichFadle (PF) series. The even and odd eigenfunctions are associated to the roots, in the complex plane, of the equations (2a) sin 2s = -2s, (2b) sin 2s = 2s. A complete study of Eqs. (2a) and (2b) may be found in [1]. Another interesting case is represented by the study of the equation (3) V4* Y2V2* = 0, where V2 = 32/3x2 + d2/dy2 and y is real. Equation (3) occurs, for example, in the study of the fluid motion in porous media [2] when Brinkman's approximation [3], [4] is used. The same equation has been used by de Socio, Gaffuri and the author [5] in the study of a tridimensional Stokes flow. If the solution of Eq. (3) is expressed in terms of PF series, the equations corresponding to (2a), (2b) are (4a) sin(i + v)/(s + v) -sin(s v)/(s v), (4b) sin(s + v)/(s + v) = sin(s v)/(s v), where v = (s2 y2)x/2. This note deals with a numerical study of Eqs. (4a), (4b) and gives a method to find all the real and complex solutions for any real value of y. Received March 29, 1983. 1980 Mathematics Subject Classification. Primary 65H05. *This work was partially supported by Ministero della Pubblica Istruzione under contract 60%, 81/00806 cap 03. ©1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page 589 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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تاریخ انتشار 2010